The towns of Franklin and Chester post their populations on signs just outside of their towns. The signs are updated once a year at the beginning of the year. During year $1$, Franklin's sign read " $\text{Franklin: Population of } 20{,}000$ ", while Chester's sign read " $\text{Chester: Population of } 25{,}000$ ". Each year, the populations grew. Specifically, Franklin's population grew by $5\%$ each year, and Chester's population grew by $500$ people each year. What is the first year in which Franklin's sign shows a larger number than Chester's sign? Year
Solution: The signs in year $2$ At the beginning of year $2$, the signs are updated. For Franklin's sign, we take the previous value, $20{,}000$, and add $5\%$ of the previous value—or just multiply the previous value by $1.05$, which amounts to the same thing: $\begin{aligned} &20{,}000+20{,}000\cdot0.05 \\\\ =&20{,}000\cdot(1+0.05) \\\\ =&20{,}000\cdot(1.05) \\\\ =& 21{,}000 \end{aligned}$ For Chester's sign, we take the previous value, $25{,}000$, and add $500$ : $\begin{aligned} &25{,}000+500 \\\\ =& 25{,}500 \end{aligned}$ The signs in year $3$ and beyond For year $3$ and beyond, we keep multiplying Franklin's population by $1.05$ and adding $500$ to Chester's population. Year Franklin Chester (Multiply by $1.05$ each year.) (Add $500$ each year.) $1$ $20{,}000$ $25{,}000$ $2$ $21{,}000$ $25{,}500$ $3$ $22{,}050$ $26{,}000$ $4$ $23{,}153$ $26{,}500$ $5$ $24{,}310$ $27{,}000$ $6$ $25{,}526$ $27{,}500$ $7$ $26{,}802$ $28{,}000$ $8$ $28{,}142$ $28{,}500$ $9$ $29{,}549$ $29{,}000$ Franklin's sign showed a larger population than Chester's sign for the first time in year number $9$. Notice: Franklin's population grows exponentially while Chester's population grows linearly.